AT-OEMO-FED1 2013 Problem 4

Let $A$ , $B$ and $C$ be three points on a line (in this order). For each circle $k$ through the points $B$ and $C$ , let $D$ be one point of intersection of the perpendicular bisector of $BC$ with the circle $k$ . Further, let $E$ be the second point of intersection of the line $AD$ with $k$ . Show that for each circle $k$ , the ratio of lengths $\overline{BE}:\overline{CE}$ is the same.